Quantitative X-ray computed tomography (XCT) diagnostics for reacting flows are developed and demonstrated in application to premixed flames in open and optically inaccessible geometries. A laboratory X-ray scanner is employed to investigate methane/air flames that were diluted with krypton as an inert radiodense tracer gas. Effects of acquisition rate and tracer gas concentration on the signal-to-noise ratio are examined. It is shown that statistically converged three-dimensional attenuation measurements can be obtained with limited impact from the tracer gas and within an acceptable acquisition time. Specific aspects of the tomographic reconstruction and the experimental procedure are examined, with particular emphasis on the quantification of experimental uncertainties. A method is developed to determine density and temperature from the X-ray attenuation measurements. These experiments are complemented by one- and multi-dimensional calculations to quantify the influence of krypton on the flame behavior. To demonstrate the merit of XCT for optically inaccessible flames, measurements of a complex flame geometry in a tubular confinement are performed. The use of a coflow to provide a uniform tracer-gas concentration is shown to improve the quantitative temperature evaluation. These measurements demonstrate the viability of XCT for flame-structure analysis and multi-dimensional temperature measurements using laboratory X-ray systems. Further opportunities for improving this diagnostic are discussed. (C) 2018 The Combustion. Institute. Published by Elsevier Inc. All rights reserved.